On Spins and Surprises

Date: 10/15/2011 at 09:29:38
From: Ty
Subject: Trying to find odds of an event happening (roulette)
In roulette, what are the odds of a single number, in this case 15,
hitting six times in nine spins?
I actually saw this happen and would like to know how rare an occurrence
this was. I can figure the odds for one spin -- there are 38 possible
outcomes on any spin of the roulette wheel, so you have a 1:38 chance of a
single number hitting on one spin -- but don't know how to do this for
multiple spins.

Date: 10/16/2011 at 23:44:30
From: Doctor Vogler
Subject: Re: Trying to find odds of an event happening (roulette)
Hi Ty,
Thanks for writing to Dr. Math.
We get a lot of questions of this form. People come to us saying things
like, "I just learned that four students in my class have the same
birthday! What is the probability of that happening?" Or, "I was playing
poker, and I got two straight flushes in a row! How rare is that?" Or, "I
flipped a coin six times yesterday, and every time it came up heads! What
are the odds of that?"
You should realize that rare events happen occasionally, and this causes
people to ask how likely it was to happen. But they don't notice all of
the many times that unlikely events *didn't* happen. So it's really not
fair to say that you are defying probability by computing the odds after
the fact.
For example, why is it that no one comes to me saying this?
"I was playing roulette the other day, and I played six times. The
first time it got a 15. Then it got 7, then 19, 1, 5, and finally 27.
What are the odds of that happening?"
Assuming you have an American roulette wheel (with 38 slots, not 37), the
probability of that event is 1/38^6. No one is astonished by that event,
and yet it is exactly the same probability as getting a 15 all six times.
Why the discrepancy in what surprises us?
If you can answer that question, then please explain to me why the same
answer doesn't apply to the question of why people are surprised when they
get a 15 six times in a row. Any answer you might give is going to
indicate that somehow the outcome of getting 15 six times in a row is
somehow more "special" in some way than getting the six numbers 15, 7, 19,
1, 5, and 27 in that order. So next you have to answer the question: What
are all of the "special" outcomes, and what is their order in degree of
"specialness"?
For example, I could calculate the probability of getting six 15's in nine
throws of the roulette wheel. But should I also include in that
calculation the probability of getting seven, eight, or nine 15's in nine
throws of the roulette wheel? Should I also include the probability of
getting six of some other number in nine throws of the roulette wheel?
After all, you would have asked the same question if you had gotten 23 six
times out of nine, right? But I'll bet that you saw that wheel spin more
than nine times. The probability of it happening in six spins out of nine
is not nearly as high as the probability of seeing six-out-of-nine in some
nine consecutive spins of a wheel that is spun fifty times.
And you probably wouldn't expect me to count, in my calculation, the
probability of the wheel giving, in those nine spins, the numbers 1, 2, 3,
4, 5, 6, 7, 8, and 9, in that order. And yet you would have asked me for
that probability if you had seen that too.
What are all of the outcomes that you might have seen that would have
caused you to be surprised and ask what the probability is?
That's what I mean when I say that rare events happen all of the time. If
you ask *before* spinning the wheel, what is the probability of getting 15
in six of the next nine spins, then my calculation would be valid, and of
course you would lose your money if you got 23 six times or got the
numbers from 1 through 9 in order. But if you see some event and then ask
what its probability is, then in some sense the answer to your question is
1 (that is, 100%) because as you already saw, it happened and there is
nothing that you can do to change that fact; the outcome is already
determined.
Similarly, if you ask *before* spinning the wheel, what is the probability
of getting the six numbers 15, 7, 19, 1, 5, and 27 in that order in the
next six spins, then the probability would be 1/38^6, and you would be
shocked out of your mind and suspect a magic trick if it actually
happened. But if you saw those numbers in that order and *then* asked what
the probability was, then someone telling you 1/38^6 would seem clearly
wrong, because it wasn't that surprising an event.
Does that make sense? I just wanted to make sure that was clear before I
did the calculation for you, because you might find the number smaller
than you expect, and I hope you now understand why.
Okay, now let's suppose you are going to make a bet with someone, and the
bet is this: I am going to spin the roulette wheel nine times. If it ends
up as a 15 for exactly six of those nine times, then you win.
I will now calculate the probability of that event happening.
On each spin of the wheel, there are 38 equiprobable outcomes, which means
that in nine spins, there are 38^9 equiprobable outcomes, each of which is
an ordered list of 9 numbers. How many of those 38^9 outcomes will have 15
exactly six times? Well, that requires six of the nine numbers to be 15,
and three of the nine to be *not* 15. There are 9-choose-6 ways to decide
which of the nine are 15 and which are not, and no matter how you choose
those, there are 37 ways to pick each of the three non-15's.
So the total number of outcomes that have 15 exactly six times is
(9-choose-6) * 37^3 = 84 * 37^3
= 4254852.
So the probability of that happening is
4254852/38^9 = 1063713/41304025315712
= 1/38830046.55928...
= 0.0000000257532526641...
Of course, if you want to change the bet to be "at least six" instead of
"exactly six," then you should add
(9-choose-7)*37^2 + (9-choose-8)*37 + (9-choose-9).
And if you want to say "some number" instead of "15," then you should
multiply by 38. But this still won't include other rare events, like
getting a 1 on every odd spin and a 2 on every even spin, or discovering
that the dealer has the same birthday as you, or the same first name, or
that this email arrives in your inbox at precisely the time (minutes and
seconds) matching the birthday (day and month) of your great Aunt Silvia.
But you can ask me those probabilities separately if you want, as long as
you realize that I won't be accepting any bets after the outcome has
already been determined.
As they say, "No more bets!"
If you have any questions about this or need more help, please write back,
and I will try to offer further suggestions.
- Doctor Vogler, The Math Forum
http://mathforum.org/dr.math/